Question

are the triangles congruent? why or why not? yes, all the angles of each of the triangles are acute. yes, they are congruent by either asa or aas. no, ∠b is not congruent to ∠q. no, the congruent sides do not correspond.

Explanation:

Step1: Calculate angle C in triangle ABC

In $\triangle ABC$, using the angle - sum property of a triangle ($\angle A+\angle B+\angle C = 180^{\circ}$), where $\angle A = 42^{\circ}$ and $\angle B=53^{\circ}$. So, $\angle C=180^{\circ}-(42^{\circ}+53^{\circ}) = 85^{\circ}$.

Step2: Analyze congruence criteria

In $\triangle ABC$ and $\triangle RMQ$, $\angle A=\angle R = 42^{\circ}$, $\angle C=\angle Q = 85^{\circ}$, and there is a pair of equal sides (the marked sides). This satisfies either the Angle - Side - Angle (ASA) or Angle - Angle - Side (AAS) congruence criteria.

Answer:

Yes, they are congruent by either ASA or AAS.